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Many video games take place in fictional universes. These universes can be mere adaptations of what really exists, such as an alternate history or a potential future, or a flight of fancy so far removed from reality as to not even attempt to be credible. Whatever the case, one way to make a fictional universe distinctive is to give it its own means of executing transactions - its own money.
Some currencies actually have their roots in the fiction of a game, such as the iconic bottlecaps from the Fallout universe. With all governments in shambles and precious metals in even shorter supply than usual, the people of the wastes trade the only shiny, easily transportable objects they have available to them - the caps from bottles of the favored drink of the Nuclear Age, Nuka-Cola.
Other games' currencies are essentially MacGuffins, objects that only have value because the game tells you they have value. This is usually the case in games with light-hearted, non-serious storylines and settings, such as rings in the Sonic universe, bolts in the universe of Ratchet and Clank, and bells in the Animal Crossing games.
Other currencies are never represented as objects in-game, and are merely words to convey the value of objects. Such a currency would be Pokebucks in the Pokemon series, or ISK in EVE Online.
Some fictional currencies are used across franchises to indicate that they exist, at least potentially, in the same universe. An example of this is all of the Capcom-made games that use Zenny as a currency.